A Subquadratic Algorithm for Road Coloring

The synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton to a single state. A coloring of edges of a directed graph is synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a synchronizing word. The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph with constant outdegree of all its vertices if the greatest common divisor of lengths of all its cycles is one. The problem was posed by Adler, Goodwyn and Weiss in 1970 and had evoked many years a noticeable interest among the specialists in the theory of graphs, deterministic automata and symbolic dynamics. A polynomial time algorithm of O(n) complexity in the most worst case and quadratic in most cases for the road coloring of the considered graph is presented below. The work is based on my recent positive solution of the road coloring problem.